ScalingStacks

Corollary 5.7 . [05BC]

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Corollary 5.7.

Let 𝔛\mathfrak{X} be a strongly nondegenerate strictly polystable formal model of XanX^{\textup{an}} over K∘K^{\circ} with associated skeleton Ξ”\Delta. Let Ο„\tau be an nn-dimensional open face of Ξ”\Delta and hh a convex function on Ο„\tau. Denote by π’ͺΒ―h∘p𝔛\overline{\mathcal{O}}^{h\circ p_{\mathfrak{X}}} the trivial bundle on pπ”›βˆ’1​(Ο„)p_{\mathfrak{X}}^{-1}(\tau) endowed with the metric given by βˆ₯1βˆ₯=eβˆ’h∘p𝔛\|1\|=e^{-h\circ p_{\mathfrak{X}}}. Then the latter is locally a semipositive metric (Definition 4.17) and

c1​(π’ͺΒ―h∘p𝔛)n=deg⁑(S)β‹…n!β‹…MA⁑(h)c_{1}\left(\overline{\mathcal{O}}^{h\circ p_{\mathfrak{X}}}\right)^{n}=\Deg(S)\cdot n!\cdot\MA(h)

on pπ”›βˆ’1​(Ο„)p_{\mathfrak{X}}^{-1}(\tau) where SS is the point in the special fibre of 𝔛\mathfrak{X} corresponding to Ο„\tau.

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